In my calculus class, we recently covered first-order, linear ODEs. Specifically, we discussed the formula for the solution of one (and its derivation): $$y=\frac{1}{u(x)}\int Q(x)u(x)dx$$ where $u(x)$ is the integrating factor defined by $$u(x) = e^{\int P(x)dx}$$ all of which can be used to solve an ODE of the form $$\frac{dy}{dx}+P(x)y=Q(x)$$ I was impressed by the beauty of this solution, and I wondered if this could be extended to other situations - specifically, a first-order linear ODE defined on $\mathbb{C}$. Is this possible?
Note: I found this post (Can a nice enough ODE always be extended to the complex plane?) which was certainly interesting, but the answer seemed a little hard to understand (I'm a high-schooler taking a second AP Calculus class); also, I'm primarily interested in the formulas I named above, and in this specific linear first-order case.
The short answer is yes.
Nevertheless, restrictions do apply: All the functions should by holomorphic and the domain simply connected (to make the integral have a unique meaning).
Say, $\Omega\subset\mathbb C$ simply connected, $P,Q\in\mathcal H(\Omega)$ and $z_0\in\Omega$. Then the IVP $$ w'=P(z)w+Q(z), \quad w(z_0)=\omega, $$ has a unique solution, namely $$ w(z)=\omega\exp\left(\int_{[z_0,z]}P(\zeta)\,d\zeta\right)+\int_{[z_0,z]}\exp\left(\int_{[z_0,\zeta]}P(\eta)\,d\eta\right)Q(\zeta)\,d\zeta. $$